Bertini irreducibility theorems over finite fields

TL;DR

This paper proves that for a geometrically irreducible subscheme in projective space over a finite field, the proportion of high-degree hypersurfaces intersecting it irreducibly approaches 100% as the degree increases, with extensions to more general settings.

Contribution

It establishes asymptotic irreducibility results for hypersurface intersections over finite fields, including variants over field extensions and more general morphisms.

Findings

Fraction of irreducible intersections tends to 1 as degree increases

Results extend to subschemes over field extensions

Applicable to more general morphisms beyond immersions

Abstract

Given a geometrically irreducible subscheme X in P^n over F_q of dimension at least 2, we prove that the fraction of degree d hypersurfaces H such that the intersection of H and X is geometrically irreducible tends to 1 as d tends to infinity. We also prove variants in which X is over an extension of F_q, and in which the immersion of X in P^n is replaced by a more general morphism.